The ternary Goldbach conjecture with primes in thin subsets
The Ternary Goldbach problem, i.e., the problem of representing all odd numbers \(N>5\) as a sum of three primes was established, for all sufficiently large odd \(N\) (say, \(N>N_0\), but with an unknown \(N_0\)), in 1937 by I. M. Vinogradov, applying his trigonometric sums-version of the circle method (of Hardy-Littlewood). Then, we may introduce further constraints on the primes \(p_i\), \(i=1,2,3\), for this equation: \[ N=p_1+p_2+p_3. \tag{\(T\)} \] In fact, the author proves \((T)\) both when the \(p_i\) lie in a constructible thin subset (TS) of the primes and when they are in arithmetic progressions (AP), say \(p_i\equiv b_i\pmod k\), with \(b_i>0\) and \((b_i,k)=1\) (He considers prime moduli \(k\)). Actually, his result for \((T)\) with primes in thin subsets follows, by a standard technique of D. Wolke, from his other AP-result. The story of \((T)\) with the \(p_i\) in TS starts with E. Wirsing, who in 1986 proved the existence of very thin subsets \(S\) of the primes, namely with \[ \left| S\cap [1,x]\right|\ll (x\log x)^{1/3} \] (here the notation \(\ll\) is another gift of I. M. Vinogradov), but \(S\) is not constructible (say, a probabilistic approach prevents us from knowing it). Then, a first explicit construction of a thin \(S\), for \((T)\) with primes in \(S\), was given by A. Balog and J. Friedlander in 1992, using Piatetski-Shapiro primes; this was generalized in 1995 by Jia to the binary Goldbach problem. However, Wolke in 1993 started the standard approach in costructing \(S\), building upon results for arithmetic progressions. We address the interested reader to the paper for a short history about these kind of results. Then, in the present paper the author gives a state-of-the-art result about \((T)\) in AP, namely it can be solved under the additional constraints \(p_i\) in AP of prime moduli \(k\) as above, for ``almost all'\', of them, precisely : \[ \forall F>0 \; \forall \, \text{prime} \, k\leq R:=N^{9/58-\varepsilon} \; \text{except} \; O(R/(\log R)^F) \; \text{of them}. \] (Here, as usual in Analytic Number Theory, \(\varepsilon>0\) is an arbitrarily small constant.) The story of \((T)\) in AP starts in 1953 with R. Ayoub, who proved a representation formula for the number of solutions, say \(J\), of it in AP as above, but with a fixed (positive integer) modulus \(k\), i.e.: \[ J=\sigma(N;k){{N^2}\over {2\log^3 N}}(1+o(1)), \] where \(\sigma(N;k)\) is the ``singular series'\', of the problem at hand, with \(\sigma(N;k)\gg 1/k^2\), see its definition in between \((1.2)\) and \((1.3)\) in the paper. Then, it came, see the paper, a flurry of papers on almost all moduli, culminating with the present. The proof of the AP result relies upon Theorem 3 (proved by Circle Method), that is a kind of Bombieri-Vinogradov result for the number of solutions \(J\) above, summed over prime moduli \(k\sim R\), i.e. \(R/2<k\leq R\): let \(\Lambda\) be the von Mangoldt function (\(\Lambda(n)=\log p\), if \(n=p^r\), \(p\) prime \& \(r\geq 1\) integer, \(\Lambda(n)=0\) otherwise), \(R\) be as above, then \(\forall A>0\), \[ \sum_{{k\sim R}\atop {k\,\text{prime}}}k\max_{(b_i,k)=1}\left| \sum_{{N=n_1+n_2+n_3}\atop {n_i\equiv b_i\;(\!\bmod k)}}\Lambda(n_1)\Lambda(n_2)\Lambda(n_3)-\sigma(N;k){{N^2}\over {32}}\right|\ll N^2(\log N)^{-A}. \]
scientific article; zbMATH DE number 5541226
scientific article